Optimal. Leaf size=367 \[ \frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}+\frac {\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 0.51, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac {\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx &=\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x^3 \left (-4 a B-\frac {1}{2} (11 b B-14 A c) x\right ) \sqrt {a+b x+c x^2} \, dx}{7 c}\\ &=-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x^2 \left (\frac {3}{2} a (11 b B-14 A c)+\frac {3}{4} \left (33 b^2 B-42 A b c-32 a B c\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{42 c^2}\\ &=\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x \left (-\frac {3}{2} a \left (33 b^2 B-42 A b c-32 a B c\right )-\frac {3}{8} \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{210 c^3}\\ &=\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^5}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^6}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{1024 c^6}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 312, normalized size = 0.85 \begin {gather*} \frac {\frac {7 \left (32 a^2 A c^3-80 a^2 b B c^2-112 a A b^2 c^2+120 a b^3 B c+42 A b^4 c-33 b^5 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2048 c^{11/2}}+\frac {x^2 (a+x (b+c x))^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{40 c^2}+\frac {(a+x (b+c x))^{3/2} \left (252 b^2 c (7 A c x-13 a B)+8 a b c^2 (343 A+333 B x)+16 a c^2 (64 a B-105 A c x)-42 b^3 c (35 A+33 B x)+1155 b^4 B\right )}{1920 c^4}+\frac {x^3 (a+x (b+c x))^{3/2} (14 A c-11 b B)}{12 c}+B x^4 (a+x (b+c x))^{3/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.82, size = 423, normalized size = 1.15 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (8192 a^3 B c^3+25312 a^2 A b c^3-6720 a^2 A c^4 x-34608 a^2 b^2 B c^2+12704 a^2 b B c^3 x-4096 a^2 B c^4 x^2-23520 a A b^3 c^2+12544 a A b^2 c^3 x-7616 a A b c^4 x^2+4480 a A c^5 x^3+21840 a b^4 B c-12096 a b^3 B c^2 x+7776 a b^2 B c^3 x^2-5056 a b B c^4 x^3+3072 a B c^5 x^4+4410 A b^5 c-2940 A b^4 c^2 x+2352 A b^3 c^3 x^2-2016 A b^2 c^4 x^3+1792 A b c^5 x^4+17920 A c^6 x^5-3465 b^6 B+2310 b^5 B c x-1848 b^4 B c^2 x^2+1584 b^3 B c^3 x^3-1408 b^2 B c^4 x^4+1280 b B c^5 x^5+15360 B c^6 x^6\right )}{107520 c^6}+\frac {\left (-128 a^3 A c^4+320 a^3 b B c^3+480 a^2 A b^2 c^3-560 a^2 b^3 B c^2-280 a A b^4 c^2+252 a b^5 B c+42 A b^6 c-33 b^7 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 843, normalized size = 2.30 \begin {gather*} \left [\frac {105 \, {\left (33 \, B b^{7} + 128 \, A a^{3} c^{4} - 160 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 280 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 42 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (15360 \, B c^{7} x^{6} - 3465 \, B b^{6} c + 1280 \, {\left (B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 32 \, {\left (256 \, B a^{3} + 791 \, A a^{2} b\right )} c^{4} - 128 \, {\left (11 \, B b^{2} c^{5} - 2 \, {\left (12 \, B a + 7 \, A b\right )} c^{6}\right )} x^{4} - 336 \, {\left (103 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} c^{3} + 16 \, {\left (99 \, B b^{3} c^{4} + 280 \, A a c^{6} - 2 \, {\left (158 \, B a b + 63 \, A b^{2}\right )} c^{5}\right )} x^{3} + 210 \, {\left (104 \, B a b^{4} + 21 \, A b^{5}\right )} c^{2} - 8 \, {\left (231 \, B b^{4} c^{3} + 8 \, {\left (64 \, B a^{2} + 119 \, A a b\right )} c^{5} - 6 \, {\left (162 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (1155 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 16 \, {\left (397 \, B a^{2} b + 392 \, A a b^{2}\right )} c^{4} - 42 \, {\left (144 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{430080 \, c^{7}}, -\frac {105 \, {\left (33 \, B b^{7} + 128 \, A a^{3} c^{4} - 160 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 280 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 42 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (15360 \, B c^{7} x^{6} - 3465 \, B b^{6} c + 1280 \, {\left (B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 32 \, {\left (256 \, B a^{3} + 791 \, A a^{2} b\right )} c^{4} - 128 \, {\left (11 \, B b^{2} c^{5} - 2 \, {\left (12 \, B a + 7 \, A b\right )} c^{6}\right )} x^{4} - 336 \, {\left (103 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} c^{3} + 16 \, {\left (99 \, B b^{3} c^{4} + 280 \, A a c^{6} - 2 \, {\left (158 \, B a b + 63 \, A b^{2}\right )} c^{5}\right )} x^{3} + 210 \, {\left (104 \, B a b^{4} + 21 \, A b^{5}\right )} c^{2} - 8 \, {\left (231 \, B b^{4} c^{3} + 8 \, {\left (64 \, B a^{2} + 119 \, A a b\right )} c^{5} - 6 \, {\left (162 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (1155 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 16 \, {\left (397 \, B a^{2} b + 392 \, A a b^{2}\right )} c^{4} - 42 \, {\left (144 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{215040 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 414, normalized size = 1.13 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B x + \frac {B b c^{5} + 14 \, A c^{6}}{c^{6}}\right )} x - \frac {11 \, B b^{2} c^{4} - 24 \, B a c^{5} - 14 \, A b c^{5}}{c^{6}}\right )} x + \frac {99 \, B b^{3} c^{3} - 316 \, B a b c^{4} - 126 \, A b^{2} c^{4} + 280 \, A a c^{5}}{c^{6}}\right )} x - \frac {231 \, B b^{4} c^{2} - 972 \, B a b^{2} c^{3} - 294 \, A b^{3} c^{3} + 512 \, B a^{2} c^{4} + 952 \, A a b c^{4}}{c^{6}}\right )} x + \frac {1155 \, B b^{5} c - 6048 \, B a b^{3} c^{2} - 1470 \, A b^{4} c^{2} + 6352 \, B a^{2} b c^{3} + 6272 \, A a b^{2} c^{3} - 3360 \, A a^{2} c^{4}}{c^{6}}\right )} x - \frac {3465 \, B b^{6} - 21840 \, B a b^{4} c - 4410 \, A b^{5} c + 34608 \, B a^{2} b^{2} c^{2} + 23520 \, A a b^{3} c^{2} - 8192 \, B a^{3} c^{3} - 25312 \, A a^{2} b c^{3}}{c^{6}}\right )} - \frac {{\left (33 \, B b^{7} - 252 \, B a b^{5} c - 42 \, A b^{6} c + 560 \, B a^{2} b^{3} c^{2} + 280 \, A a b^{4} c^{2} - 320 \, B a^{3} b c^{3} - 480 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 872, normalized size = 2.38 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,x^{4}}{7 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,x^{3}}{6 c}-\frac {11 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b \,x^{3}}{84 c^{2}}+\frac {A \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {15 A \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {7}{2}}}+\frac {35 A a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}-\frac {21 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {11}{2}}}-\frac {5 B \,a^{3} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {7}{2}}}+\frac {35 B \,a^{2} b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {63 B a \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {11}{2}}}+\frac {33 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {13}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} x}{16 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2} x}{32 c^{3}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} x}{256 c^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b \,x^{2}}{20 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b x}{32 c^{3}}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3} x}{64 c^{4}}-\frac {4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,x^{2}}{35 c^{2}}-\frac {33 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5} x}{512 c^{5}}+\frac {33 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} x^{2}}{280 c^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} b}{32 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{3}}{64 c^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a x}{8 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{512 c^{5}}+\frac {21 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} x}{160 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b^{2}}{64 c^{4}}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{4}}{128 c^{5}}+\frac {111 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b x}{560 c^{3}}-\frac {33 \sqrt {c \,x^{2}+b x +a}\, B \,b^{6}}{1024 c^{6}}-\frac {33 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} x}{320 c^{4}}+\frac {49 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a b}{240 c^{3}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{64 c^{4}}+\frac {8 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,a^{2}}{105 c^{3}}-\frac {39 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,b^{2}}{160 c^{4}}+\frac {11 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{128 c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 992, normalized size = 2.70 \begin {gather*} \frac {8\,B\,a^3\,\sqrt {c\,x^2+b\,x+a}}{105\,c^3}-\frac {33\,B\,b^6\,\sqrt {c\,x^2+b\,x+a}}{1024\,c^6}+\frac {A\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{6\,c}+\frac {B\,x^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{7\,c}+\frac {33\,B\,b^7\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{2048\,c^{13/2}}+\frac {A\,a\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{2\,c}-\frac {3\,A\,b\,\left (\frac {7\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )}{4\,c}-\frac {5\,B\,a^3\,b\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{32\,c^{7/2}}-\frac {63\,B\,a\,b^5\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{512\,c^{11/2}}+\frac {35\,B\,a^2\,b^3\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{128\,c^{9/2}}+\frac {13\,B\,a\,b^4\,\sqrt {c\,x^2+b\,x+a}}{64\,c^5}-\frac {4\,B\,a\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{35\,c^2}-\frac {11\,B\,b\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{84\,c^2}-\frac {33\,B\,b^3\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{320\,c^4}+\frac {11\,B\,b^5\,x\,\sqrt {c\,x^2+b\,x+a}}{512\,c^5}-\frac {103\,B\,a^2\,b^2\,\sqrt {c\,x^2+b\,x+a}}{320\,c^4}+\frac {8\,B\,a^2\,x^2\,\sqrt {c\,x^2+b\,x+a}}{105\,c^2}+\frac {33\,B\,b^2\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{280\,c^3}+\frac {11\,B\,b^4\,x^2\,\sqrt {c\,x^2+b\,x+a}}{128\,c^4}-\frac {39\,B\,a\,b^2\,x^2\,\sqrt {c\,x^2+b\,x+a}}{160\,c^3}+\frac {111\,B\,a\,b\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{560\,c^3}-\frac {269\,B\,a^2\,b\,x\,\sqrt {c\,x^2+b\,x+a}}{3360\,c^3}-\frac {3\,B\,a\,b^3\,x\,\sqrt {c\,x^2+b\,x+a}}{320\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (A + B x\right ) \sqrt {a + b x + c x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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